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This author-recorded argument by Plasma AI prepares the degree-four case of Problem 114. It gives compact coordinates, an exact identity reducing the length inequality to one signed integral, an explicit uniform deficit at sufficiently small curvature, and computer-assisted bounds on both collision strata. The quartic inequality remains unproved on the regular complement.

Exact first and second variations reduce possible maximum-principle and angular-gap convexity arguments to moment inequalities for an individual quartic. Those inequalities remain open. Observations in other degrees motivate the route but provide no all-degree conclusion. The cubic argument integrates inverse-derivative densities against those of $z^3-c$. Its quartic analogue fails near $z^4-1$, and the section on the rotated comparison replaces it.

By Eremenko and Hayman's Lemma 4 and Lemma 5, some length-maximizing monic quartic has all critical values on the unit circle. These are external source premises at the reading depth their result pages record. Their existence assertion suffices for an upper bound for every quartic; it does not put every quartic into the family below or settle uniqueness. Translation centers the polynomial, and $p(z)\mapsto e^{-4i\theta}p(e^{i\theta}z)$ preserves length and monicity.

A compact parameter set

L186 records this elementary parameter reduction and its unicritical comparison, with a complete author-recorded proof. The numbered formulas below give the coordinates used in the later deficit estimates. L186 proves these coordinate statements, not a length comparison on the generic or two collision families.

Write $p(z)=z^4+az^2+bz+c$ and let $u,v,w$ be its critical points, counted with multiplicity. Then

$$u+v+w=0,\qquad p'(z)=4(z-u)(z-v)(z-w),\qquad p(u)-p(v)=w(u-v)^3. \tag{1}$$

For the last identity integrate $p'$ from $v$ to $u$, writing $z=(u+v)/2+t$. The odd part vanishes, and the integral is $-(u+v)(u-v)^3$. The identity also holds when critical points coincide.

Let $M=\max(|u|,|v|,|w|)$ and label $|u|=M$. If $|v-w|\ge M/2$, then $|p(v)-p(w)|\ge M^4/8$. Otherwise $|v|,|w|>M/4$ and $|u-v|=|3u/2-(v-w)/2|>5M/4$, so $|p(u)-p(v)|>125M^4/256$. Consequently

$$\max_{i,j}|p(u_i)-p(u_j)|\ge M^4/8. \tag{2}$$

If all critical values have modulus one, (2) gives $M\le2$. Comparing coefficients in $p'$ now gives

$$|a|\le6M^2\le24,\qquad |b|\le4M^3\le32, \qquad |c|\le1+M^4+|a|M^2+|b|M\le177. \tag{3}$$

The centered coefficient set with all critical values on the circle is closed: along coefficient convergence the multiset of roots of $p'$ converges, and their polynomial values converge. It is bounded by (3), hence compact. More generally, critical values in $|z|\le K$ give $M^4\le16K$ and a coefficient bound depending only on $K$. This also supplies compactness when critical values are moved along fixed bounded arcs. No classification theorem for the critical-value map is needed for these bounds.

The generic two-real-parameter chart

Assume first that the critical points and the critical values are each pairwise distinct. Choose a critical point of largest modulus, and rotate so that it equals $s>0$. Write the other two as $st,-s(1+t)$, where

$$|t|\le1,\qquad |1+t|\le1.$$

Define

$$F_t(z)=z^4-2(1+t+t^2)z^2+4t(1+t)z.$$

Then $p(sz)=s^4F_t(z)+c$. Put $f_1=F_t(1)$, $f_2=F_t(t)$ and $f_3=F_t(-1-t)$. The three distinct critical values on a circle are not collinear. Thus the $f_i$ have a unique circumcenter $C(t)$ and positive circumradius $R(t)$. They can be obtained from the two real linear equations

$$2\operatorname{Re}\bigl((C-f_1)\overline{(f_i-f_1)}\bigr) =|f_i-f_1|^2,\qquad i=2,3.$$

The unit-circle condition forces $s^4R(t)=1$ and $c=-s^4C(t)$. Conversely these choices give unit critical values. Every member of the generic class is therefore, up to the stated isometries, of the form

$$p_t(z)=s^4\bigl(F_t(z/s)-C(t)\bigr),\qquad s=R(t)^{-1/4}. \tag{4}$$

The parameter domain is the part of the closed lens $|t|,|1+t|\le1$ where the three $f_i$ are noncollinear and the three critical points are distinct. Relabeling critical points can give several parameters for the same polynomial; injectivity is unnecessary for coverage. The chart itself is singular at excluded parameters. Compactness of the coefficient set does not give bounded derivatives of (4) there.

The missing strata are explicit

There are three additional cases, with complex conjugation allowed to identify the two signs below, since it preserves length.

A double critical point. The roots of $p'$ are $r,r,-2r$ with $r\ne0$. Rotate so that $r>0$ and put $h=r^4$. The polynomial is

$$p(z)=z^4-6\sqrt h\,z^2+8h^{3/4}z+ \frac{21h}{2}\ \pm i\sqrt{1-\frac{729h^2}{4}}, \qquad 0<h\le\frac2{27}. \tag{5}$$

Indeed its critical values are $c+3h$ and $c-24h$, whose difference is $27h$. Requiring both moduli to equal one gives (5). This family has a double critical point and a distinct simple one throughout $h>0$.

Distinct critical points with two equal critical values. By (1), if $u\ne v$ but $p(u)=p(v)$, the third critical point is zero. Thus the roots are $r,-r,0$. Rotate so that $r>0$ and again put $h=r^4$. The family is

$$p(z)=z^4-2\sqrt h\,z^2+\frac h2 \ \pm i\sqrt{1-\frac{h^2}{4}},\qquad 0<h\le2. \tag{6}$$

Its critical values are $c-h,c-h,c$, so the unit-circle condition gives (6). These are collisions of values at distinct points, and are not covered by considering repeated roots of $p'$ alone.

A triple critical point. Centering forces it to be zero, so $p(z)=z^4+c$ with $|c|=1$. Rotation makes this $z^4-1$.

These cases exhaust the complement of (4). Three distinct collinear critical values cannot lie on the unit circle. If all critical values equal $v_0$, each distinct critical point of multiplicity $m$ is a root of $p-v_0$ of multiplicity $m+1$. Their total multiplicity is three plus the number of distinct critical points, at most four. Thus there is just one critical point, the last case above.

The length integral and the unicritical comparison

Away from the finitely many critical values, inverse branches give

$$\ell(p)=\int_0^{2\pi} \sum_{p(z)=e^{i\theta}}\frac1{|p'(z)|}\,d\theta. \tag{7}$$

For $p(z)=z^4-1$ this density is $|1+e^{i\theta}|^{-3/4}$, so

$$\ell(z^4-1)=2^{1/4}B\!\left(\frac12,\frac18\right). \tag{8}$$

A critical point of multiplicity $m$ in $p'$ gives a local inverse-derivative singularity of order $|\theta-\theta_0|^{-m/(m+1)}$, by the first nonzero term of the Taylor expansion of $p-p(z_0)$. Thus the pointwise singular orders are $1/2$, $2/3$ and $3/4$. Two distinct critical points with the same value contribute two singular terms, rather than a higher ramification order. All are integrable for a fixed polynomial. Their leading constants need not be uniform as critical points merge.

There is nonetheless an explicit uniform bound for a whole short arc. The proof of Eremenko--Hayman Lemma 4, using its preceding projection estimate and Cartan covering lemma, gives for an arc $K$ of the unit circle of length $0<\delta<\pi/6$

$$\int_K\sum_{p(z)=e^{i\theta}}\frac{d\theta}{|p'(z)|} \le64e(\delta/2)^{1/4}. \tag{9}$$

This source estimate is uniform over all monic quartics. Thus short arcs around the at most three critical values have a controlled total contribution even during collisions. The bound is coarse and tends to zero independently of the polynomial; it does not compare the small length deficit near the proposed extremizer.

For the single-critical-point class, L186's unicritical comparison has the following elementary calculation. Put $\Phi(c)=\ell(z^4-c)$ for real $c\ge0$. For $c<1$, expanding the two factors of $|1+ce^{i\theta}|^{-3/4}$ and integrating gives

$$\Phi(c)=2\pi\sum_{j\ge0}\binom{-3/8}{j}^{\!2}c^{2j}.$$

It is strictly increasing on $[0,1)$. Dominated convergence at $c=1$ follows from $|c+e^{i\theta}|^2=(1-c)^2+c|1+e^{i\theta}|^2$ for $c\ge1/2$. For $c>1$, $\Phi(c)=c^{-3/4}\Phi(1/c)<\Phi(1)$. Rotating the constant term therefore shows that $\ell(z^4+c)\le\ell(z^4-1)$, with equality exactly when $|c|=1$.

The rotated comparison

Let $p$ be a centered monic quartic and $\lambda>0$, and put

$$F(\zeta)=\lambda^{-4}\bigl(p(\lambda\zeta)-p(0)\bigr),\qquad \Gamma=\{\sigma:|\sigma+\lambda^{-4}p(0)|=\lambda^{-4}\},\qquad G_F(\sigma)=\sum_{F(\zeta)=\sigma}\frac1{|F'(\zeta)|}.$$

The lemniscate of $p$ is $\lambda F^{-1}(\Gamma)$, so (7) applied to $F$ gives $\ell(p)=\lambda\int_\Gamma G_F|d\sigma|$. For every $\sigma_0\in\Gamma$,

$$\ell(p)-\ell(z^4-1)=\lambda\,\tilde I(F,\Gamma),\qquad \tilde I(F,\Gamma)=\int_\Gamma\Bigl(G_F(\sigma)-|\sigma-\sigma_0|^{-3/4}\Bigr) |d\sigma|. \tag{10}$$

Indeed $|\sigma-\sigma_0|^{-3/4}$ is the inverse-derivative density of $\zeta^4+\sigma_0$. Since $\sigma_0$ lies on $\Gamma$, whose radius is $R=\lambda^{-4}$, the set $\{\zeta:\zeta^4+\sigma_0\in\Gamma\}$ is $R^{1/4}$ times a rotated lemniscate of $z^4-1$. Its length is $R^{1/4}\ell(z^4-1)=\lambda^{-1}\ell(z^4-1)$. Both integrals are finite, so $\tilde I$ does not depend on the choice of $\sigma_0\in\Gamma$. Identity (10) loses nothing: on the nonunicritical part of the compact family the strict length comparison says exactly that $\tilde I<0$. The unicritical members with unit critical value have $\tilde I=0$. In the chart (4), $\lambda=s=R(t)^{-1/4}$ and $F=F_t$.

Why the cubic comparison fails here

The cubic argument compares with the polynomial of the same constant term. Here that is $z^4+p(0)$, whose density over $\Gamma$ is $|\sigma|^{-3/4}$:

$$\lambda I(F,\Gamma)=\ell(p)-\ell(z^4+p(0)),\qquad I(F,\Gamma)=\int_\Gamma\bigl(G_F(\sigma)-|\sigma|^{-3/4}\bigr)|d\sigma|.$$

Thus $\lambda\tilde I=\lambda I-(\ell_0-\Phi(|p(0)|))$, where $\ell_0=\ell(z^4-1)$. Since $\Phi\le\ell_0$, the sufficient condition $I<0$ implies $\tilde I<0$, but it is strictly stronger, and near the extremizer the two differ at the same order. As $r\to1$, $\Phi(1)-\Phi(r)\sim K_\Phi|1-r|^{1/4}$ with

$$K_\Phi=\int_{-\infty}^{\infty}\bigl(|u|^{-3/4}-(1+u^2)^{-3/8}\bigr)\,du =\frac{8\,\Gamma(1/2)\,\Gamma(7/8)}{\Gamma(3/8)}\approx6.519 .$$

Also $1-|p(0)|=(R-|C|)/R$ for the center $C$ of $\Gamma$. When $\Gamma$ flattens to a line $L$, $R-|C|$ tends to plus or minus the distance from $0$ to $L$. Hence, along any approach to $z^4-1$ in the family,

$$\frac{\ell_0-\Phi(|p(0)|)}{\lambda}\to K_\Phi\operatorname{dist}(0,L)^{1/4}, \qquad I(F,\Gamma)\to\tilde I_\infty(F,L)+K_\Phi\operatorname{dist}(0,L)^{1/4},$$

with $\tilde I_\infty$ the line integral defined below. On the real segment of the collinear locus, and at the line ends of both collision strata, $L$ passes through $0$ and the extra term vanishes. On the arch it does not. The sign reversal is proved without numerical quadrature in L176. Put $Y=\sqrt{9/4+3\sqrt3/2}$ and $F_y=z^4+(2y^2-3/2)z^2-(1+4y^2)z$. For $y>Y$ let $\Gamma_y$ be the circle through its three critical values. As $y\downarrow Y$, these circles flatten to $L=-(6+3\sqrt3)+i\mathbb R$, their radii $R_y$ tend to infinity, and the associated monic quartics tend to $z^4+1$. The claim proves

$$\lim_{y\downarrow Y}I(F_y,\Gamma_y)>\frac{31}{20}.$$

Its geometric proof bounds the harmonic preimage of $L$ using its eight ends and its two finite edges. Thus the comparison with the same constant term fails arbitrarily close to the unicritical extremizer even when all three critical points and all three unit critical values are distinct. This is an author-recorded obstruction to that comparison, not to the quartic inequality. Identity (10) repairs the reference; its integral is strictly negative sufficiently close to the extremizer, as shown below.

Floating-point evaluations also suggest failure over a larger part of the arch. At $t\approx-0.3+0.2857i$ they give $\operatorname{dist}(0,L)\approx0.1885$, $\tilde I_\infty\approx-2.291$ and a limit of $I$ near $+2.003$. Slightly off the arch, at $s\approx0.086$, they give $I\approx+2.002$ and $\tilde I\approx-2.293$. These values are exploratory and do not certify a parameter interval.

Optimizing fixed weights at the critical values does not repair a pointwise comparison. The author-recorded L182 constructs regular unit-critical-value quartics arbitrarily close to $z^4-i$ for which no weights $\alpha_j\ge0$, $\sum_j\alpha_j=1$, satisfy

$$\sum_{p(z)=\zeta}\frac1{|p'(z)|} \le\sum_{j=1}^3\alpha_j|\zeta-v_j|^{-3/4} \qquad(\zeta\in\mathbf T\setminus\{v_1,v_2,v_3\}).$$

The weights may depend on $p$, but must remain fixed around the circle. Each reference integrates to $\ell_0$. The obstruction comes from the limit $F=z^4-2z^2$ on the real line: at the two targets $T,-T$, the sum of its densities has an extra positive term $T^{-7/4}/8$, whereas every such mixture has cancellation at that order. A strict two-point gap persists on large circles and under separation of the critical values. This rules out the entire fixed convex-mixture comparison, including equal weights; it leaves integrated comparisons open.

The parameter space and its degenerations

The substitutions $t\mapsto-1-t$, which exchanges two critical points, and $t\mapsto\bar t$, which conjugates the polynomial, preserve length. So the lens may be cut to

$$D=\{t:-\tfrac12\le\operatorname{Re}t\le0,\ \operatorname{Im}t\ge0,\ |1+t|\le1\}.$$

By (1) and the multiplicity count, its only shapes with fewer than three distinct critical values are $t=0$ and $t=-\tfrac12$. At $t=0$ the critical points are $1,0,-1$ with values $-1,0,-1$. At $t=-\tfrac12$ they are $1,-\tfrac12,-\tfrac12$ with values $-\tfrac32,\tfrac3{16},\tfrac3{16}$. In general (1) gives the exact differences

$$f_1-f_2=(t-1)^3(t+1),\qquad f_1-f_3=t(2+t)^3,\qquad f_2-f_3=(1+2t)^3$$

for the values $f_1,f_2,f_3$ at $1,t,-1-t$, so collisions are visible with relative accuracy. Over $t=0$ the circles through the two values form a pencil. It is exactly family (6), with radius $R=1/h$ and $\lambda=h^{1/4}$. Over $t=-\tfrac12$ the pencil is family (5), with $R=1/(16h)$ and $\lambda=2h^{1/4}$ after the rotation $z\mapsto-z$. Each pencil also contains its line, the limit $h\to0$.

Elsewhere the circle through the three values is unique unless they are collinear. Floating-point sign tracking, not validated, puts the collinear locus in $D$ on the segment $[-\tfrac12,0]$, where $F_t$ is real, and on an arch. The arch leaves $t=-\tfrac12$ at angle $\pi/3$ and meets $|1+t|=1$ near $t\approx-0.105+0.428i$. Approaching the locus, $R\to\infty$ and $\lambda\to0$, so the polynomials tend to rotations of $z^4-1$. The computations below never rely on this description: they treat curvature intervals containing $0$ directly.

Let $P$ be the set of pairs $(F_t,\Gamma)$ with $t\in D$ and $\Gamma$ a circle or line through all critical values of $F_t$. Here is a concrete compact topology, including both pencils. By (2), since the largest critical-point modulus is one, the diameter of the critical values is at least $1/8$. Every circle in $P$ therefore has radius at least $1/16$. Represent its curve by the point $f_1$, a unit tangent $e$, and signed curvature $\kappa\in[-16,16]$, with $\kappa=0$ for a line. The incidence conditions are the closed equations

$$\operatorname{Im}((f_j-f_1)\bar e) =\frac\kappa2|f_j-f_1|^2,\qquad j=2,3.$$

They define a closed subset of $D\times\mathbf T\times[-16,16]$; identifying the two orientations $(e,\kappa)$ and $(-e,-\kappa)$ gives $P$. Thus $P$ is compact. Each member with a circle is the scaled picture of a polynomial of the family; members with a line are limits only. For a line $L$ through $\sigma_0$ put

$$\tilde I_\infty(F,L)=\int_L\bigl(G_F(\sigma)-|\sigma-\sigma_0|^{-3/4}\bigr) |d\sigma| .$$

It converges absolutely by (b) below and does not depend on $\sigma_0\in L$. It equals the renormalized length $\lim_{r\to\infty}\bigl(|F^{-1}(L)\cap D_r|-8r\bigr)$ of the zero set of a harmonic polynomial of degree four, measured against eight rays. The two truncations differ by $O(1/r)$, since $|F(z)-\sigma_0|=|z|^4(1+O(|z|^{-2}))$. A rotation of the variable writes $F^{-1}(L)=\{\operatorname{Re}P=0\}$ with $P$ monic, centered and with a real constant term. The author-recorded quantitative estimate L179 gives

$$\tilde I_\infty(F,L)\le -\frac{\max(|a|^{1/2},|b|^{1/3})}{2500}\le-\frac1{2500}$$

on the normalized shape family. The last inequality follows from $4\le2|a|+|b|$ at a critical point of modulus one. L179 makes the projection argument in Fryntov and Nazarov's Lemma 1 quantitative in degree four; that source lemma is recorded in the proof pointer of their local maximality result. The estimate does not require the other critical values to lie on $L$.

Uniform estimates through the degenerations

The following facts apply to centered monic quartics $F(z)=z^4+az^2+bz$ and circles or lines $\Gamma$. On the compact shape family, cluster windows and the far-field estimate provide uniform control through collisions and flattening. The single-point weights in (c) are used only where their denominators stay separated from zero.

(a) Crofton windows. $F^{-1}(\Gamma)$ is a real algebraic curve of degree at most eight. Restricted to a line, its equation is a real polynomial of degree at most eight, which vanishes identically for at most finitely many lines, so almost every line meets the curve in at most eight points. Crofton's formula gives $|F^{-1}(\Gamma)\cap D(w,r)|\le8\pi r$ for every disc. Let $w$ be a critical point of multiplicity $m$ with value $f$, and write $F(w+t)-f=\sum_{k>m}c_kt^k$. If $\eta=|c_{m+1}|r^{m+1}-\sum_{k>m+1}|c_k|r^k>0$, then for $|\sigma-f|<\eta$ Rouché's theorem gives $F-\sigma$ exactly $m+1$ zeros in $D(w,r)$. An arc $W\subset\Gamma$ within distance $\eta$ of $f$ therefore contributes at most $8\pi r$ per such disc, plus $|W|$ times a bound for $1/|F'|$ at the remaining zeros. When critical points merge, one disc around the cluster serves. When the values of distinct critical points merge, each point keeps its own disc. For a fixed limiting cluster and a fixed disc whose boundary contains no root, the Rouché margin stays positive on a sufficiently small parameter neighborhood. The margin is not uniform if the disc radius is also allowed to shrink with the separation of two critical points. Inside windows this local bound replaces (9).

(b) A product identity and the far field. If $z_1,\dots,z_4$ solve $F(z)=\sigma$ and $f_1,f_2,f_3$ are the critical values with multiplicity, the resultant of $F-\sigma$ and $F'$ gives

$$\prod_{k=1}^{4}F'(z_k)=256\prod_{j=1}^{3}(f_j-\sigma). \tag{11}$$

By the arithmetic-geometric mean inequality, $G_F(\sigma)\ge\prod_j|\sigma-f_j|^{-1/4}$. Put $y_k=\log\bigl(4\prod_j|\sigma-f_j|^{1/4}/|F'(z_k)|\bigr)$. Then (11) gives $\sum_ky_k=0$, hence

$$G_F(\sigma)=\prod_j|\sigma-f_j|^{-1/4}\,(1+\Delta),\qquad \Delta=\frac14\sum_k\bigl(e^{y_k}-1-y_k\bigr)\ge0 .$$

With $\varepsilon_k=1/z_k$ one has

$$y_k=\operatorname{Re}\psi(\varepsilon_k) -\frac14\sum_m\operatorname{Re}\psi(\varepsilon_m),\qquad \psi(\varepsilon)=\tfrac34\operatorname{Log}(1+a\varepsilon^2+b\varepsilon^3) -\operatorname{Log}(1+\tfrac a2\varepsilon^2+\tfrac b4\varepsilon^3) =\tfrac a4\varepsilon^2+\tfrac b2\varepsilon^3+O(\varepsilon^4).$$

The power sums $\sum_k\varepsilon_k^2=b^2/\sigma^2+2a/\sigma$ and $\sum_k\varepsilon_k^3=b^3/\sigma^3+3ab/\sigma^2$ are $O(1/\sigma)$. So $y_k=O(|\sigma|^{-1/2})$ and $\Delta=O(1/|\sigma|)$, with explicit constants. Consequently

$$G_F(\sigma)-|\sigma-\sigma_0|^{-3/4}=|\sigma-\sigma_0|^{-3/4} \Bigl(\prod_j\Bigl|\frac{\sigma-f_j}{\sigma-\sigma_0}\Bigr|^{-1/4}(1+\Delta) -1\Bigr), \tag{12}$$

and beyond an explicit radius $T$ the absolute value of the bracket is at most $K/|\sigma-\sigma_0|$. The constant $K$ depends only on $|a|$, $|b|$, the critical values, $\sigma_0$ and $T$.

(c) Singular weights. If $F-f=(z-w)^{m+1}Q(z)$, every zero $z$ of $F-\sigma$ satisfies

$$|F'(z)|=\Bigl(\frac{|\sigma-f|}{|Q(z)|}\Bigr)^{m/(m+1)} \bigl|(m+1)Q(z)+(z-w)Q'(z)\bigr|. \tag{13}$$

At a fixed multiplicity pattern, with the other critical points separated, this gives a bounded weighted density near $f$. During a merger $Q(w)$ can tend to zero; then (a) supplies a cluster window instead of asserting a uniform bound from this formula alone.

(d) Flattening. Take a base point $\sigma_*\in\Gamma$, the tangent direction $e^{i\alpha}$ there, and the signed curvature $\kappa$ of $\Gamma$. Then

$$\sigma(x)=\sigma_*+\frac{e^{i\alpha}x}{1-i\kappa x/2},\qquad |d\sigma|=\frac{dx}{1+\kappa^2x^2/4},\qquad |\sigma-\sigma_*|^{-3/4}|d\sigma| =\frac{|x|^{-3/4}\,dx}{(1+\kappa^2x^2/4)^{5/8}},$$

and $\kappa=0$ is the line. With $u=1/x$, the arc near the antipode of $\sigma_*$ carries the integrand $(u^2+\kappa^2/4)^{-5/8}$ times the bracket of (12). By (b) this is at most $K|u|^{-1/4}$, uniformly in $\kappa$.

Here is the continuity argument with the collision quantifiers explicit. Fix a limiting member of $P$, and surround each distinct root above each of its critical values by disjoint small discs. On their boundaries $|F-\sigma|$ has a positive margin when $\sigma$ is sufficiently close to the corresponding value. Rouché's theorem preserves the root counts under nearby parameters. The whole preimage of each such value window is therefore contained in those discs, and (a) bounds its length by $8\pi$ times the sum of their radii, uniformly nearby. This applies whether the merging roots are critical points or distinct points with merging values. The reference window contributes $O(\eta^{1/4})$ uniformly, since circle radii are at least $1/16$. Outside these windows and a large disc in the value plane, simple inverse branches give uniform convergence.

For the remaining tail (b) gives a uniform $O(|\sigma|^{-7/4})$ bound on the signed density difference, with $\sigma_0=f_1$. A circle has at most $2\pi r$ of length in a disc of radius $r$, so summing over dyadic annuli bounds the absolute tail by $O(T^{-3/4})$, uniformly also for lines and arbitrarily large circles. Let the parameter converge first, then shrink the windows and enlarge $T$. Thus $\tilde I$ extends continuously to $P$ without an assertion of uniform Puiseux constants at merging points.

The author-recorded L180 provides a fully explicit, though very coarse, version near the line part: for its stated constant $\kappa_0>0$,

$$|\kappa|\le\kappa_0\quad\Longrightarrow\quad \tilde I(F,\Gamma)\le-\frac1{5000},\qquad \ell(p)\le\ell_0-\frac{\lambda}{5000}. \tag{17}$$

Here $|\kappa|=R^{-1}=\lambda^4$. The proof compares a circle with its tangent line for the same $F$. It uses a uniform far-field constant, fourth-root coverings of the whole preimage of each critical-value window, and the resultant (11) on the remaining compact interval. It covers every direction of approach, including both collision strata, without a separation assumption on critical points or critical values. Its cutoff is not a practical mesh size for a numerical cover.

Validated bounds on the collision strata

On each stratum the scaled polynomial is fixed, $F_5=z^4-\tfrac32z^2-z$ for family (5) and $F_6=z^4-2z^2$ for family (6), and $\Gamma$ runs over the pencil of circles through its two critical values together with the line. A ball-arithmetic program proves the author-recorded claim L181, with its method and results retained in the collision-strata evidence, namely $\tilde I(F_5,\Gamma)<-1$ and $\tilde I(F_6,\Gamma)<-1$ for every member of both pencils; its largest upper bounds are less than $-1.021$ and $-1.077$, respectively. It applies (a) to (d), certifies every root it uses by Rouché's theorem, and bisects boxes of the pencil parameter until each bound is below $-1$. With (10), every polynomial of family (5) has length below $\ell_0-2h^{1/4}$, and every polynomial of family (6) has length below $\ell_0-h^{1/4}$, for all admissible $h>0$. This is the uniform signed deficit along each collision stratum, up to and including the approach to $z^4-1$. It is author-recorded and computer-assisted, without independent review.

Continuity on $P$ also supplies a neighborhood of the two compact pencils on which $\tilde I<-1/2$. Together with the neighborhood of the line part, this gives a justified uniform treatment of every degeneration. The pencil neighborhoods have no explicit numerical radius here; (17) does give one for the line part. A validated finite cover still needs explicit overlap bounds for the pencils before it can omit their neighborhoods and cover the remaining compact regular set.

Exploratory landscape

Floating-point quadrature, not validated, used tanh-sinh rules on the arcs between critical values and numpy roots; the landscape probe of the collision-strata evidence recomputes the values below. It sampled a grid of $D$, polar grids around both special shapes, the collinear locus and both pencils, and found $\tilde I\le-1.69$ everywhere. The largest value, about $-1.6914$, occurs at the smallest circle of family (5). Along the real segment $\tilde I_\infty$ runs from $-1.71$ near $t=-\tfrac12$ to $-2.39$ near $t=0$. Along the arch it runs from the same value at $t=-\tfrac12$, where both branches meet the line of family (5), to $-2.65$ near the lens boundary. On family (5) $\tilde I$ lies in $[-1.711,-1.691]$, and on family (6) in $[-2.408,-2.396]$. If these samples are representative, $\ell(p)\le\ell_0-1.69\lambda$ on the whole family.

A root-free main term

For three distinct critical values put $k=(f_3-f_1)/(f_3-f_2)$ and $A=k(f_1-f_2)$; in the chart (4), identity (1) gives $k=t(2+t)^3/(1+2t)^3$ and $A=k(t-1)^3(t+1)$. Then $\sigma(x)=f_1+Ax/(1-kx)$, for $x\in\mathbb R\cup\{\infty\}$, runs over the circle or line through the values, with $f_1$, $f_3$ and $f_2$ at $x=0$, $1$ and $\infty$. The radius is $R=|A|/(2|\operatorname{Im}k|)$, and

$$\frac{|\sigma-f_1|}{|\sigma-f_2|}=|k|\,|x|,\qquad \frac{|\sigma-f_1|}{|\sigma-f_3|}=\frac{|1-k|\,|x|}{|x-1|}.$$

With $\sigma_0=f_1$, (12) therefore gives $\tilde I=|A|^{1/4}\bigl(J_0(k)+J_\Delta\bigr)$, where

$$J_0(k)=\int_{-\infty}^{\infty}\frac{|k(1-k)|^{1/4}|x|^{1/2}|x-1|^{-1/4}-1} {|x|^{3/4}\,|1-kx|^{5/4}}\,dx,\qquad J_\Delta=\int_{-\infty}^{\infty}\frac{|k(1-k)|^{1/4}|x|^{1/2}|x-1|^{-1/4}} {|x|^{3/4}\,|1-kx|^{5/4}}\,\Delta\,dx\ge0 .$$

The main term $J_0$ depends on the shape only through $k$ and needs no roots; the subtracted weight integrates to $\ell_0/(2|\operatorname{Im}k|)^{1/4}$. Its integrand vanishes to first order at $x=1/k$, which keeps it integrable when $\Gamma$ is a line. Floating-point values, recomputed by the landscape probe, give $J_0=-3.16$ and $J_\Delta=+1.49$ at $t=-0.295+0.505i$, and $J_0=-0.98$ and $J_\Delta=+0.51$ at $t=-0.45+0.1i$ near the double point. The inequality on generic shapes thus amounts to $J_\Delta<-J_0$, and certified roots, with their interval slack, enter only the smaller term.

A stronger comparison using squared inverse derivatives

For a regular target value define

$$G_{2,F}(\sigma)=2\left(\sum_{F(z)=\sigma}|F'(z)|^{-2}\right)^{1/2}.$$

Cauchy--Schwarz gives $G_F\le G_{2,F}$. Thus, with the same reference point as in (10),

$$I_2(F,\Gamma)=\int_\Gamma \left(G_{2,F}(\sigma)-|\sigma-\sigma_0|^{-3/4}\right)|d\sigma| \le0 \tag{19}$$

would suffice for the length bound. For a regular quartic the Cauchy--Schwarz step is strict on an arc next to any critical value: two inverse derivatives become unbounded and the other two remain bounded. Consequently (19), even without a strict sign, gives (H) on the regular family. Equality in the pointwise step does hold for a unicritical polynomial. Inequality (19) is stronger than the target and remains unproved.

There is an exact negative line limit at one collision shape. For $F=z^4-2z^2$, solving first for $z^2$ gives, at every regular complex value, the root-free formula

$$G_{2,F}(\sigma) =\frac{\bigl((1+|\sigma|+|\sigma+1|)/2\bigr)^{1/4}} {|\sigma|^{1/2}|\sigma+1|^{1/2}}.$$

Indeed, with $q^2=1+\sigma$, the squared density is $(|1+q|^{-1}+|1-q|^{-1})/(2|1+\sigma|)$, and $(|1+q|+|1-q|)^2=2+2|1+\sigma|+2|\sigma|$. On the real line, taking $\sigma_0=0$ therefore gives

$$G_{2,F}(\sigma)= \begin{cases} \sigma^{-1/2}(1+\sigma)^{-1/4},&\sigma>0,\\ (-\sigma)^{-1/2}(1+\sigma)^{-1/2},&-1<\sigma<0,\\ (-\sigma)^{-1/4}(-1-\sigma)^{-1/2},&\sigma<-1. \end{cases}$$

The middle interval contributes $\pi$. The two exterior integrals have the same finite part, because

$$\int_0^B x^{-1/2}(1+x)^{-1/4}\,dx =\frac{4\sqrt B}{(1+B)^{1/4}} -\int_0^B x^{-1/2}(1+x)^{-5/4}\,dx.$$

Subtracting the reference and letting $B\to\infty$ therefore gives

$$I_2(F,\mathbb R)=\pi-2B(1/2,3/4)<\pi-4<0, \tag{20}$$

where $B(1/2,3/4)>\int_0^1t^{-1/2}\,dt=2$. This is an analytic bound at the line member only, not on the full pencil.

For that pencil the root-free formula reduces the proposed inequality to one scalar trigonometric integral. If $\beta\in(0,\pi/2]$ is the tangent-chord angle and $R=(2\sin\beta)^{-1}$, then

$$I_2(F,\Gamma)=R^{1/4}\bigl(\mathcal J(\beta)-\ell_0\bigr),\qquad \mathcal J(\beta)=\int_0^\pi \frac{\bigl(\sin\beta+\sin\theta+|\sin(\theta+\beta)|\bigr)^{1/4}} {\bigl(\sin\theta\,|\sin(\theta+\beta)|\bigr)^{1/2}}\,d\theta.$$

This follows by writing the two chord lengths as $2R\sin\theta$ and $2R|\sin(\theta+\beta)|$. The inequality $\mathcal J(\beta)\le\ell_0$ on the whole interval is not proved here. Thus even this explicit formula leaves a substantive one-dimensional comparison.

The same continuity mechanism as for $\tilde I$ applies to $I_2$: $G_{2,F}\le2G_F$ controls cluster windows, and the inverse expansion $\Psi'(w)=1+a/(4w^2)+b/(2w^3)+O(w^{-4})$ at infinity shows that $G_{2,F}(\sigma)=|\sigma|^{-3/4}(1+O(|\sigma|^{-1}))$. The terms of orders $w^{-2}$ and $w^{-3}$ cancel when averaging their four rotations before taking the square root. On the compact shape set the remainder is uniform. The signed tail is consequently $O(|\sigma|^{-7/4})$, and (20) gives a neighborhood of this line member with negative $I_2$.

The landscape probe gives, in floating point, $I_2$ approximately $-0.843,-1.241,-0.883,-0.757,-0.959,-0.867$ at $t=-0.317+0.282i,-0.05+0.107i,-0.25+0.6i,-0.4+0.05i, -1/2+i\sqrt3/2,-0.3+0.2857i$, respectively. These are six individual observations, not coverage of a region. The next analytic test for this comparison is the sign of $I_2$ on the remaining line shapes and the collision pencils. At sixty sampled positive angles on each pencil, the probe finds $I_2$ between $-0.664$ and $-0.644$ for the double-point pencil and between $-1.719$ and $-1.650$ for the equal-values pencil. These ranges are floating-point observations, with no interval coverage. The established deficits for $G_F$ do not imply deficits for this larger density.

A possible maximum principle

Formula (4) defines a polynomial $p_t$ of the family for every $t\in\mathbb C$ whose critical values are distinct and not collinear, not only for $t$ in the lens, and its length $\ell(t)=\ell(p_t)$ depends only on the similarity class of the critical points $1,t,-1-t$. Relabeling them acts on $t$ by the Möbius maps $t\mapsto-1-t$, $1/t$, $-(1+t)/t$, $-1/(1+t)$ and $-t/(1+t)$, and conjugation acts by $t\mapsto\bar t$; $\ell$ is invariant under all of them. It is real-analytic off the collinear locus (the real axis and the images of the arch) and the special shapes, it is bounded, and by Eremenko--Hayman Lemma 4 it tends to $\ell_0$ at every point of the locus other than the special shapes, where its limit depends on the direction of approach.

Floating-point evaluations found the discrete Laplacian of $\ell$, with step $0.003$, positive at all 129 points of a $15\times15$ grid of $D$ whose stencils avoid the locus, and at further points closer to it; the landscape probe rechecks five of them. This suggests the following conjecture: $\ell$ is subharmonic on $\mathbb C$ minus the collinear locus and the special shapes.

The conjecture would give (H) without a two-dimensional cover. On each component of the complement of the locus, $\ell$ would be bounded and subharmonic. The special shapes are finitely many boundary points, which the extended maximum principle may ignore, and $\ell\to\ell_0$ at every other boundary point. Hence $\ell\le\ell_0$, and $\ell<\ell_0$ by the strong maximum principle, since $\ell$ is not constant.

A weaker statement holds. Write $\ell=(2|\operatorname{Im}k|/|A|)^{1/4}\ell_w$, where $\ell_w$ is the length of $F_t^{-1}(\Gamma)$ and $k$, $A$ are as in the previous section. With $M_t(x)=f_1+Ax/(1-kx)$, the curve $F_t^{-1}(\Gamma)$ is $\Phi_t^{-1}(\mathbb R\cup\{\infty\})$ for the rational map $\Phi_t=M_t^{-1}\circ F_t$ of degree four, which depends holomorphically on $t$. Its critical values are $0$, $1$ and $\infty$, the images of $f_1$, $f_3$ and $f_2$, which do not move, and $1/k$, the image of the triple critical point at infinity, which stays off the real line away from the locus. Hence, on each component, for every real $x\ne0,1$ the four preimages of $x$, and $1/\Phi_t'$ at them, depend holomorphically on $t$, and

$$\ell_w(t)=\int_{-\infty}^{\infty}\sum_{\Phi_t(w)=x}\frac{dx}{|\Phi_t'(w)|}$$

is an integral of sums of moduli of holomorphic functions, hence log-subharmonic; the integrability is locally uniform by (a) and (c). The missing factor $(2|\operatorname{Im}k|/|A|)^{1/4}$ has a superharmonic logarithm. For the reference it exactly cancels, since $(2|\operatorname{Im}k|)^{1/4}\int|x|^{-3/4}|1-kx|^{-5/4}\,dx=\ell_0$. The conjecture therefore asks that $\ell_w$ be at least as log-subharmonic as the corresponding reference integral $\ell_0/s$.

That comparison has an exact pointwise form. If $u=\int\sum_j|h_j(x,t)|\,dx$ with each $h_j$ holomorphic in $t$, then $\Delta\log u$ is the variance of $\partial_t\log h_j(x,t)$ under the probability measure proportional to $|h_j(x,t)|\,dx$ on the pairs $(x,j)$, where $\operatorname{Var}(g)=\int|g|^2-\bigl|\int g\bigr|^2$. For a centered monic quartic $p=z^4+az^2+bz+c$ let $V_p$ be the polynomial part of $p^2/p'$, so that

$$V_p'(z)=\frac54z^4+\frac{9a}8z^2+\frac{7b}8z+\frac{a^2}{16}+\frac c2,$$

and let $\mu_p$ be arclength on $|p|=1$ divided by $\ell(p)$. Then, away from the locus and the special shapes,

$$(\operatorname{Im}k)^2\,\Delta_k\log\ell =\tfrac14\bigl(\operatorname{Var}_{\mu_p}(V_p')-1\bigr),\qquad p=p_t, \tag{14}$$

where $\Delta_k$ is the Laplacian in the local coordinate $k$. So $\log\ell$ is subharmonic exactly when $\operatorname{Var}_{\mu_p}(V_p')\ge1$ for every polynomial $p$ of the family, a condition on each polynomial separately. The author-recorded L178 derives the same local variation from a polynomial differential equation, and also computes the second derivative needed for the angular-gap test below. Neither derivation proves the sign of the variance expression.

Proof of (14). Put $\beta=1/k$. It is a local coordinate off the special shapes, since $k'(t)=2(2+t)^2(1-t)^2/(1+2t)^4$ vanishes only at $t=-2$ and $t=1$, the images of $t=-\frac12$ under the relabelings, and $(\operatorname{Im}k)^2\Delta_k=(\operatorname{Im}\beta)^2\Delta_\beta$. Since $\Phi_t$ takes the value $\beta$ only at $w=\infty$, where it is totally ramified, $\Psi=1/(\Phi_t-\beta)$ is a polynomial of degree four in $w$. After a holomorphic change of the coordinate $w$, which changes $\log\ell_w$ by a harmonic function, $\Psi$ is monic and centered. Its critical values $-1/\beta$, $1/(1-\beta)$ and $0$, the images of $0$, $1$ and $\infty$, each satisfy $dv/d\beta=v^2$. The curve is the preimage under $\Psi$ of the circle $\{1/(x-\beta):x\in\mathbb R\cup\{\infty\}\}$, which has center $i/(2\operatorname{Im}\beta)$ and passes through $0$. For real $x\ne0,1$ the four solutions $w_j$ of $\Psi(w)=1/(x-\beta)$ are holomorphic in $\beta$, and the length is $\int\sum_j|h_j|\,dx$ with $h_j=-1/((x-\beta)^2\Psi'(w_j))$.

Write a dot for $\partial_\beta$ at fixed $w$. Differentiating $\Psi(w_j)=1/(x-\beta)$ gives $\dot\Psi+\Psi'\dot w_j=\Psi^2$ at $w_j$. The polynomial $\dot\Psi$ has degree at most two and equals $\dot v=v^2$ at each critical point, so it is the remainder of $\Psi^2$ on division by $\Psi'$, and $\dot w_j=V(w_j)$ for the quotient $V$. Differentiating $\Psi'V=\Psi^2-\dot\Psi$ in $w$ then gives $\partial_\beta\log h_j=V'(w_j)$, so $\Delta_\beta\log\ell_w$ is the variance of $V'$ under normalized arclength on the curve. In the variable $z$ of the monic polynomial $p$ of the family, $\Psi=\alpha(p-\gamma)$ with $w=sz$, $s^4=\alpha$, $\gamma$ a critical value of $p$ and $-\alpha\gamma=i/(2\operatorname{Im}\beta)$, the center of the circle. Division gives $V(w)=\alpha s\bigl(V_p(z)-\gamma z/2\bigr)$, so that variance is $\operatorname{Var}_{\mu_p}(V_p')/(2\operatorname{Im}\beta)^2$. Finally $\ell=(2|\operatorname{Im}\beta|)^{1/4}\ell_w$ in this normalization and $\Delta_\beta\log|\operatorname{Im}\beta|=-(\operatorname{Im}\beta)^{-2}$, which gives (14).

The proof also gives the first variation. The complex derivative $\partial_\beta\log\ell_w$ is half the mean of $V'$, and the constant $-\alpha\gamma/4$ cancels the derivative of $\frac14\log|\operatorname{Im}\beta|$, so $\partial_\beta\log\ell=\frac\alpha2\int V_p'\,d\mu_p$, in which the constant term of $V_p'$ matters. Hence $|\operatorname{Im}k|\,|\nabla_k\log\ell|=\frac12\bigl|\int V_p'\,d\mu_p\bigr|$, and with (14)

$$(\operatorname{Im}k)^2\,\frac{\Delta_k\ell}{\ell} =\frac14\Bigl(\int|V_p'|^2\,d\mu_p-1\Bigr). \tag{15}$$

So $\ell$ itself is subharmonic exactly when $\int_{|p|=1}|V_p'|^2\,|dz|\ge\ell(p)$ for every polynomial of the family, and the critical points of $\ell$ are the shapes with $\int V_p'\,d\mu_p=0$. A weaker condition already gives (H): that $\int|V_p'|^2\,d\mu_p>1$ whenever $\int V_p'\,d\mu_p=0$. It excludes interior local maxima of $\ell$. By the full second variation in L178, a condition implied by it also suffices. At a member with $\int V_p'\,d\mu_p=0$ the Hessian of $\log\ell$ in the local disc of L178 has the eigenvalues $\frac12\bigl(\int|V_p'|^2,d\mu_p-1\bigr)\pm \bigl|\mathbb E_{\mu_p}W+\frac12\mathbb E_{\mu_p}(V_p')^2\bigr|$, with $W$ as in the section on the gaps, and it is enough that the larger one be positive at every such member. At the boundary of each component, $\ell$ has limit superior at most $\ell_0$: at the locus by Lemma 4, and at the special shapes because, by the compactness after (3), the polynomials $p_t$ accumulate there only at unicritical polynomials and members of the strata, whose lengths are at most $\ell_0$ by the validated bounds, and length is continuous by Lemma 4.

Equality holds in the limit at the locus. For $p=z^4+c$ with $|c|=1$, $V_p'=\frac54p-\frac34c$, and $p$ pushes $\mu_p$ forward to the measure proportional to $|\zeta-c|^{-3/4}|d\zeta|$ on $\mathbf T$. With $\zeta=-ce^{2i\phi}$ this weight is proportional to $|\cos\phi|^{-3/4}$ and $\zeta-c=-2c\cos\phi\,e^{i\phi}$, so the mean $1/5$ of $\cos^2\phi$ gives $\zeta$ the variance $\frac45-\frac4{25}=\frac{16}{25}$ and $V_p'$ the variance $1$; the mean of $V_p'$ is $\frac54\cdot\frac35c-\frac34c=0$. As $t$ approaches a point of the locus other than a special shape, $p_t$ tends to such a polynomial. Formally, to first order in $s$, the expansion $\ell=\ell_0+s\tilde I$ with $s$ proportional to $|\operatorname{Im}k|^{1/4}|A|^{-1/4}$ gives $\operatorname{Var}_{\mu_p}(V_p')-1\approx-\frac34\tilde I_\infty s/\ell_0$. This is positive by Lemma 1 of Fryntov and Nazarov, but it tends to $0$, so the condition has no uniform margin near the locus. In floating point the ratio of $\operatorname{Var}_{\mu_p}(V_p')-1$ to this prediction decreases to $1$ from above as $t$ approaches the segment at $-0.25$ (from $1.45$ to $1.05$) or the arch on either side (to $1.03$).

In the hyperbolic metric $|dk|/|\operatorname{Im}k|$ of a component, with $\Delta_h=(\operatorname{Im}k)^2\Delta_k$, identity (15) reads

$$\Delta_h(\ell_0-\ell)+\tfrac14(\ell_0-\ell) =\tfrac14\Bigl(\ell_0-\int_{|p|=1}|V_p'|^2\,|dz|\Bigr).$$

Near the locus $\ell_0-\ell$ is asymptotic to $\lambda|\tilde I_\infty|$, and $\lambda=(2|\operatorname{Im}k|/|A|)^{1/4}$, with $A$ holomorphic, satisfies $\Delta_h\lambda=-\frac3{16}\lambda+o(\lambda)$ there. If $\Delta_h(\ell_0-\ell)\le-\frac3{16}(\ell_0-\ell)$ held on every component, equivalently $\int|V_p'|^2\,d\mu_p-1\ge\frac34(\ell_0/\ell-1)$, comparison with $|\operatorname{Im}k|^{1/4}$, a positive solution of $\Delta_h\phi=-\frac3{16}\phi$, would give (H). Floating-point quadrature indicates failure of this comparison, which is exact to first order at the locus. At $t=-0.14545+0.39692i$, the computation, stable to seven digits when the step is halved, gives $\int|V_p'|^2\,d\mu_p-1=0.084335$ against $\frac34(\ell_0/\ell-1)=0.084405$. There also $|\int V_p'\,d\mu_p|>\frac12(\ell_0/\ell-1)$, so the hyperbolic gradient of $\log(\ell_0-\ell)$ has length above $\frac14$, the limit of that of $\log\lambda$ at the locus. On 100 shapes of a $12\times14$ grid of $D$ the ratio of the two sides of the comparison lies between $0.9992$ and $1.235$, while $\int|V_p'|^2\,d\mu_p-1\ge0.060$. Comparisons through (15) with an eigenvalue below $\frac3{16}$, such as the subharmonicity of $\ell$ itself, keep slack at first order at the locus and remain open.

Floating-point values of $\operatorname{Var}_{\mu_p}(V_p')$ at $t=-0.317+0.282i$, $-0.05+0.107i$, $-0.25+0.6i$ and $-0.4+0.05i$ are $1.089$, $1.295$, $1.200$ and $1.192$. They match the discrete Laplacians of $\log\ell$ through (14) to three or four digits, and the discrete $(\operatorname{Im}k)^2\Delta_k\log\ell$ lies between $0.022$ and $0.074$ at the 82 points of a $12\times12$ grid of $D$ whose stencils avoid the locus. The smallest values occur near the arch. At the same four shapes $\frac12|\int V_p'\,d\mu_p|$ and $\frac14(\int|V_p'|^2\,d\mu_p-1)$ match the discrete hyperbolic gradient of $\log\ell$ and the discrete $(\operatorname{Im}k)^2\Delta_k\ell/\ell$ to the same accuracy. The landscape probe rechecks all three identities at the first and third shapes. So the stronger statement that $\log\ell$ is subharmonic may also hold.

Newton's method on $\int V_p'\,d\mu_p=0$ from the best points of a grid of $D$ finds the equilateral shape $t=-\frac12+\frac{\sqrt3}2i$, the polynomial $z^4+bz$, where the mean vanishes by its threefold symmetry and $\int|V_p'|^2\,d\mu_p=1.2268$. Started below the arch, it runs to the real axis, which belongs to the locus and where the mean tends to $0$, without converging; no other critical point of $\ell$ in $D$ was found. The probe rechecks the mean and mean square at the equilateral shape.

An explicit coefficient bound is available. The author-recorded L183 puts $S=\ell(p)/(2\pi)$, $m=\mathbb E_{\mu_p}Q$, and

$$A=\frac{21a^2}{512}-\frac{9c}{32},\qquad D=\frac{25}{16}+\frac{|a|^2}{1024}+\frac{|b|^2}{256} +\frac{25|ab|^2}{16384},$$

and proves $S>1$ and

$$\operatorname{Var}_{\mu_p}(Q) \ge \frac DS+\frac{|A-m|^2}{S-1}. \tag{18}$$

Its proof transfers arclength to the boundary norm of $g=\sqrt{\Psi'}$, where $p(\Psi(w))=w^4$ outside the unit circle. Polynomial division fixes five Laurent modes of $(Q-m)g$; orthogonality to $g$ and Cauchy--Schwarz control the remaining modes. At a stationary member, $m=0$ in (18). In particular, a regular local maximum must satisfy $D/S+|A|^2/(S-1)\le1$. This is an explicit necessary constraint, not a classification of the possible maxima.

At $p=z^4+c$, $|c|=1$, the right side of (18) is $25/(16S_0)+81/(1024(S_0-1))<1$, where $S_0=\ell_0/(2\pi)$, although the actual variance is one. L183 proves the strict loss by showing that the remaining Laurent modes are not proportional. The coefficient bound therefore cannot supply the variance sign in a full neighborhood of the unicritical limit: its right side is continuous there, by (9) and the boundedness of the polynomial weights. More modes can improve the bound, but no tail sign or uniform cutoff is supplied by (18).

No simple splitting of the variance carries the inequality. The variance of $p$ itself under $\mu_p$ stays above $\frac{16}{25}$ at the sampled shapes (minimum $0.674$), but the other terms of $V_p'$ lower the variance. Both the variance of the conditional mean of $V_p'$ given $p$ and the Cauchy--Schwarz bound $|\operatorname{Cov}(V_p',p)|^2/\operatorname{Var}(p)$ fall below $1$, to $0.959$ and $0.948$.

Nor can $\ell_0/4$ bound every petal separately, as the author-recorded L184 proves. Each component of $\{|p|<1\}$, a petal, is mapped by $p$ conformally onto the unit disc, and the four petal boundaries make up the lemniscate. For an exact obstruction take $p=z^4+4r^3z$ with $r=3^{-1/4}$. Its critical points $c_j=r\exp((2j+1)\pi i/3)$ have unit critical values. For $0\le s<1$, $p(sc_j)=(4s-s^4)p(c_j)/3$ lies in the unit disc, so all three critical points are on the boundary of the petal containing zero. That boundary has length at least the perimeter $3\sqrt3r=3^{5/4}$ of their equilateral triangle. But

$$\frac{\ell_0}{4} =2^{-3/4}\int_0^{\pi/2}(\sin u)^{-3/4}\,du \le2^{1/4}\pi<3^{5/4},$$

using $\sin u\ge2u/\pi$ and $\pi<22/7$. Thus the petal bound is rigorously false. The floating-point calculation gives that central petal length $1.468\,\ell_0/4$ and variance of $V_p'$ equal to $0.733$; the other three have length $0.553\,\ell_0/4$ and variance $1.629$ each. The variance values are numerical observations only. A proof using petal maps must account for the four petals collectively.

The same computation applies to cubics, whose family is the pencil of circles through the two critical values; there the length depends on $\beta$ only through $\theta=\arg\beta$, the tangent-chord angle of the circle. For the length $f(\theta)$ it gives $\sin^2\theta\,(\log f)''=\frac14\operatorname{Var}_{\mu_p}(V_p')-\frac13$, with $V_p'=\frac43z^3+\frac{10a}9z+\frac{2b}3$ for $p=z^3+az+b$, and the variance is exactly $\frac43$ for $z^3+c$. Floating-point values of both sides agree to four digits and are positive at seven angles in $(0,\frac\pi2]$, two of which the probe rechecks, so the length of the cubic pencil appears to be log-convex in $\theta$. No proof of $\operatorname{Var}_{\mu_p}(V_p')\ge1$ for quartics is known.

The local computation uses the degree only through the normalization exponent. In degree $n$, the analogous polynomial differential equation $\partial_\tau q=R_q$, with $R_q$ the remainder of $q^2$ modulo $q'$, moves simple critical values by $v\mapsto v/(1-\tau v)$. The corresponding formal version of (14) is $(\operatorname{Im}\beta)^2\Delta_\beta\log\ell= \frac14\operatorname{Var}_{\mu_p}(V_p')-\frac1n$, with $V_p$ the polynomial part of $p^2/p'$ as before, and the variance is exactly $4/n$ for $z^n+c$ with $|c|=1$. If this identity, global continuation with bounded normalized coefficients, and approximation of collision configurations are justified in that degree, the inequality $\operatorname{Var}_{\mu_p}(V_p')\ge4/n$ would give the analogous maximum-principle route. Those additional global steps are not established here; no properness or openness assertion about the general critical-value map is used in the quartic argument. At the configurations sampled in floating point, $n\operatorname{Var}/4$ lies between $1.05$ and $1.15$ in degree three, $1.16$ and $1.23$ in degree four, $1.23$ and $1.36$ in degree five, including eight branches reached from random polynomials, and $1.39$ and $1.43$ in degree six; the probe rechecks one configuration in each of degrees five and six. The ratio tends to $1$ only at the unicritical limit.

Convexity in the gaps of the critical values

A statement stronger than the conjectures of the previous section is natural in other coordinates. After an affine change of variable the monic polynomial of the family is $T_\beta\circ\Phi_t$, with $T_\beta(x)=-e^{i\varphi}(x-\bar\beta)/(x-\beta)$ for a real $\varphi$, so its critical values are $T_\beta(0)$, $T_\beta(1)$ and $T_\beta(\infty)$. The three arcs of $\mathbf T$ between them therefore have lengths $2\pi\omega_1$, $2\pi\omega_2$ and $2\pi\omega_3$, where $\omega_i$ are the harmonic measures of $(-\infty,0)$, $(0,1)$ and $(1,\infty)$ seen from whichever of $\beta$ and $\bar\beta$ lies in the upper half-plane. The locus is the preimage of the real line under $k$, a rational map of degree four whose critical values $0$, $1$ and $\infty$ are real. So $k$ maps each component of the complement of the locus conformally onto a half-plane, and on each component $\ell$ is a function $\Lambda$ of the gaps $(\omega_1,\omega_2,\omega_3)$ on the open triangle $\omega_i>0$, $\sum\omega_i=1$. Its vertices correspond to the locus and its edges to the special shapes, where two critical values merge.

Since the $\omega_i$ are harmonic, convexity of $\Lambda$ or $\log\Lambda$ in the gaps would imply that $\ell$ or $\log\ell$ is subharmonic. It would also give (H) directly, with no maximum principle. On every segment through an interior point a convex function lies below the larger of its limits superior at the two ends, which are at most $\ell_0$ as shown above, and equality at one interior point would make $\Lambda$ constant. Central differences of $\log\Lambda$ in the gaps, with steps down to $1/80$ of the smallest gap and Richardson extrapolation, give positive definite Hessians at all 44 shapes of a $9\times9$ grid of $D$ away from the locus. The smallest extrapolated eigenvalue, $0.08$, occurs next to an edge; coarser plain differences show small negative values there, which vanish as the step shrinks. The probe rechecks the arcs at two shapes and the convexity at two interior shapes. Near an edge the convexity needs the strata lengths to be convex along the edge; both are log-convex in the gap at 41 angles, and the probe rechecks nine. Near a vertex, say that of $(0,1)$, let $\rho$ be the sum of the two small gaps and $x=\omega_3/\rho$. Then $\ell=\ell_0+\rho^{1/4}K(x)+o(\rho^{1/4})$ with $K(x)=(2\pi(1-x)/(x|A|))^{1/4}\tilde I_\infty$, because $s^4=2|\operatorname{Im}k|/|A|$, $\beta\approx x$ and $|\operatorname{Im}\beta|\approx\pi\rho x(1-x)$. A function $\rho^{1/4}K(x)$ is convex in the gaps exactly when $K<0$ and $K^4$ is concave in $x$, since its Hessian determinant is a negative multiple of $(K^4)''$. Here the tangential and normal second derivatives have the same order, so this is a genuine condition; subharmonicity needs neither. Computed directly as $(\ell-\ell_0)/\rho^{1/4}$ at $\rho=10^{-6}$, $K^4$ is concave at 47 points of $(0,1)$ on each of the four branches; the probe rechecks five points of one branch.

There is an exact local criterion for this conjecture. Use the notation of L178: $Q=V_p'$, $R_p=p^2-p'V_p=r_2z^2+r_1z+r_0$, $m=\mathbb E_{\mu_p}Q$, and

$$W=V_pQ'+\frac{9r_2}{8}z^2+\frac{7r_1}{8}z +\frac{ar_2}{8}+\frac{r_0}{2}.$$

Order the three critical values $v_1,v_2,v_3$ counterclockwise, put $\delta_j=v_{j+1}-v_j$, $\epsilon_j=v_{j+1}^2-v_j^2$, and define

$$J=\frac1\pi \begin{pmatrix} \operatorname{Im}\delta_1&\operatorname{Re}\delta_1\\ \operatorname{Im}\delta_2&\operatorname{Re}\delta_2 \end{pmatrix},\qquad J^{\mathsf T}g=(\operatorname{Re}m,-\operatorname{Im}m)^{\mathsf T}.$$

Then the Hessian of $\log\ell$ in these two angular gaps is positive semidefinite precisely when

$$\operatorname{Var}_{\mu_p}(Q)-1\ \ge\ 2\left|\mathbb E_{\mu_p}W+ \frac{\mathbb E_{\mu_p}Q^2-m^2}{2} +\frac{i}{\pi}\sum_{j=1}^2g_j\epsilon_j\right|. \tag{16}$$

This is a stronger unresolved inequality than the trace condition (14). It includes the acceleration of the angular-gap coordinates, which cannot be omitted when testing their convexity. A positive trace alone does not establish (16). Its proof is analytic and local, with no numerical differentiation or global critical-value continuation premise.

One can pose the analogous conjecture in other degrees wherever a local branch with the prescribed critical values has been constructed: on every branch, $\log\ell$ is a convex function of the gaps between the distinct critical values on $\mathbf T$, whatever their multiplicities. A global deduction by induction on the number of values would additionally require continuation of those branches and control of their boundary limits; those assertions in arbitrary degree are not established here. Floating-point Hessians of $\log\ell$ in the gaps are positive definite at every configuration sampled: in degree three along the cubic pencil, in degree four on the grid above, in degree five at nine configurations on the branch of $z^5+bz$ and at eight on branches reached from random polynomials, and in degree six at four such random branches and one configuration on the branch of $z^6+bz$. The smallest eigenvalues lie between $0.12$ and $0.52$ in degrees five and six. The probe rechecks one configuration on the branch of $z^n+bz$ in each of degrees five and six. This is a conjecture with numerical support only.

From the strict bound to the theorem and its equality case

Let (H) be the statement that $\tilde I(F,\Gamma)<0$ for every member of $P$ whose $\Gamma$ is a circle. The following reduction is author-recorded and uses Lemmas 4 and 5 of Eremenko and Hayman as source premises.

By (10), (H) says that every centered monic quartic with all critical values on the unit circle $\mathbf T$, other than $z^4+c$, is shorter than $z^4-1$. Indeed such a polynomial, rotated and scaled, has a shape in the lens, the symmetries move it into $D$ without changing its length, and its scaled unit circle passes through all its critical values.

Upper bound. By Lemmas 4 and 5 some maximizer $p^*$ has all critical values on $\mathbf T$. If $p^*$ is not of the form $(z-z_0)^4+c$, (H) gives $\ell(p^*)<\ell_0$, which contradicts maximality. So $p^*$ is unicritical, and $\ell(p)\le\ell(p^*)=\ell_0$ for every monic quartic $p$.

Equality. Under the upper bound just proved, any $p$ with $\ell(p)=\ell_0$ is a global, hence local, maximizer. The source-dependent author-recorded deduction L177 shows that every local maximizing quartic has all critical values on $\mathbf T$. Thus (H) makes $p$ unicritical, and the unicritical comparison forces $p(z)=(z-z_0)^4+c$ with $|c|=1$. These are precisely the translated and length-preserving rotated copies of $z^4-1$.

L177 uses the source's local deformation for an arbitrary maximizer, not only the existential statement of Lemma 5. If its length were constant, equality in the holomorphic mean-value inequality would force the derivative of the straightening map to be one on the lemniscate and then throughout the complement of the deformation support. The principal normalization at infinity would make the map the identity there, contradicting the moved critical value. This is a deduction from the source construction, not an independent audit of it.

There is also a continuation argument using that same source construction without its strictness deduction. For a nonunicritical maximizer $p$ there are at least two distinct critical values, by the multiplicity count in the section on the strata. If they all lie on $\mathbf T$, (H) gives a contradiction. If a value $v$ lies off $\mathbf T$, the deformation in the proof of Lemma 5 applies to the extremal polynomial $p$. It moves $v$, and every critical point with value $v$, to $v+\lambda$ for small $|\lambda|$, keeps the other critical values fixed, and keeps the length maximal. Join $v$ to a point $\zeta\in\mathbf T$ by a simple arc $\gamma$ that meets $\mathbf T$ only at $\zeta$ and avoids the other critical values. The set of $\tau\in\gamma\setminus\{\zeta\}$ such that some maximizer has critical values $\tau$ and the others is nonempty and open by the deformation. It is closed: centered maximizers whose critical values stay bounded have bounded coefficients by the remark after (3), a subsequence converges, and its limit is a maximizer by Lemma 4 with the limiting critical values. Connectedness gives every $\tau$, and the same limit gives a maximizer with critical values $\zeta$ and the others. Repeat for each value off $\mathbf T$, choosing distinct endpoints. Critical points may merge in these limits, but values never do, since each arc avoids the other values. The result is a maximizer with all critical values on $\mathbf T$ and at least two distinct values, which (H) excludes.

So (H) implies that $\ell(p)\le\ell(z^4-1)$ for every monic quartic, with equality exactly for the rotations and translations of $z^4-1$. The alternative continuation argument follows the cubic one in L144; that claim's standing does not cover it. (H) is proved here only on the two pencils and qualitative neighborhoods of the degenerations, so no inequality for all quartics and no equality classification is claimed.

Remaining obligations

By the previous section, the inequality for all quartics and its equality case reduce to (H). The conjectured maximum principle above would prove (H) directly, and so would the inequality $\operatorname{Var}_{\mu_p}(V_p')\ge1$ for each polynomial of the family, by (14), the stationary-point test after (15) in its Hessian form, or the convexity of the length in the gaps between the critical values. Without them, the stronger integrated comparison (19) is another sufficient target, but its uniform line and collision-pencil signs remain open. For a direct cover of the original density, the two pencils are done and three steps remain.

1. Neighborhoods of the strata. Cover $\rho\le\rho_0$ in the blow-up coordinates $t=t_*+\rho e^{i\varphi}$ around $t_*=0$ and $t_*=-\tfrac12$, with $\rho=0$ included. Near $t=0$ each critical point of the merging pair keeps its own Crofton disc, and $f_1-f_3=t(2+t)^3$. Near $t=-\tfrac12$ one disc surrounds the cluster, and $f_2-f_3=(1+2t)^3=8\rho^3e^{3i\varphi}$. After division by these factors the circle data are smooth in $(\rho,\varphi)$. Identity (13) is applied relative to each moving critical point, with every root certified in the frame of its nearest critical point so that box widths do not swamp the singular scale.
2. The collinear locus. Estimate (17) gives an explicit analytic cutoff. A practical cover needs a much larger certified strip, using curvature intervals containing $0$ in the parametrization of (d), with the far field bounded by (b).
3. The regular complement in $D$, by ordinary boxes. Only this step involves no degeneration.

In all three steps, plain interval boxes in the shape variable lose the cancellation between the two terms of $J_0$ and inside $J_\Delta$, so mean-value forms in the shape variable, with the root-free term $J_0$ handled separately, are needed for boxes of practical size.